Healing times for circular wounds on plane and spherical bone surfaces

A mathematical model is developed for the rate of healing of a circular wound in a spherical “skull”. The motivation for this model is based on experimental studies of the “critical size defect” (CSD) in animal models; this has been defined as the smallest intraosseous wound that does not heal by bone formation during the lifetime of the animal [1]. For practical purposes, this timescale can usually be taken as one year. In [2], the definition was further extended to a defect which has less than ten percent bony regeneration during the lifetime of the animal. CSDs can “heal” by fibrous connective tissue formation, but since this is not bone, it does not have the properties (strength, etc.) that a completely healed defect would. Earlier models of bone wound healing [3,4] have focused on the existence (or not) of a CSD based on a steady-state analysis, so the time development of the wound was not addressed. In this paper, the time development of a circular cylindrical wound is discussed from a general point of view. An integro-differential equation is derived for the radial contraction rate of the wound in terms of the wound radius and parameters related to the postulated healing mechanisms. This equation includes the effect of the curvature of the (spherical) skull, since it is clear from the experimental evidence that the size of the CSD increases monotonically with the size of the calvaria. Certain special cases for a planar wound are highlighted to illustrate the qualitative features of the model, in particular, the dependence of the wound healing time on the initial wound size and the thickness of the healing rim.     

Hyperbolastic modeling of wound healing

A new mathematical model for wound healing is introduced and applied to three sets of experimental data. The model is easy to implement but can accommodate a wide range of factors affecting the wound healing process. The data sets represent the areas of trace elements, diabetic wounds, growth factors, and nutrition within the field of wound healing. The model produces an explicit function accurately representing the time course of healing wounds from a given data set. Such a function is used to study variations in the healing velocity among different types of wounds and at different stages in the healing process. A new multivariable model of wound healing capable of analyzing the effects of several variables on accelerating the wound healing process is also introduced. Such a model can help to formulate appropriate strategies to treat wounds. It also would enable us to evaluate the efficacy of different treatment modalities during the inflammatory, proliferative, and tissue remodeling phases.     

Local meshless method for PDEs arising from models of wound healing

A meshless collocation procedure is proposed for one- and two-dimensional partial differential equations arising from modeling of wound healing processes (Sherratt and Murray, 1991). Main motivation of this choice is its straightforward application in higher dimensions for both regular and irregular domains on various nodal points distributions. In the case of numerical solution of convection-dominated wound healing PDE models, a stencil based upwind stabilization technique is coupled with the local meshless method to counter instabilities of the computed solution. To assess efficacy, efficiency and accuracy of the proposed method on regular and irregular domains, numerical approximations of different wound healing models are obtained and validated against the exact solution and medically tested healing time duration.     

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0,1) or (1,2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.     

Biological implications of a discrete mathematical model for collagen deposition and alignment in dermal wound repair

We deveiop a novel mathematical model for collagen depositionand alignment during dermal wound healing. We focus on the interactionsbetween fibroblasts, modelled as discrete entities, and a continuousextracellular matrix composed of collagen and a fibrin basedblood clot. There are four basic interactions assumed in themodel: fibroblasts orient the collagen matrix, fibroblasts produceand degrade collagen and fibrin and the matrix directs the fibroblastsand determines the speed of the cells. Several factors whichinfluence the alignment of collagen are examined and relatedto current anti-scarring therapies using transforming growthfactor ß. The most influential of these factors arecell speed and, more importantly for wound healing, the influxof fibroblasts from surrounding tissue.     

Models of shrinking clusters with applications to epidermal wound healing

Models of growing clusters, such as the Eden model and Diffusion Limited Aggregation (DLA), have been widely used to describe a variety of natural growth processes. In this paper, we develop models of shrinking clusters which we use to model epidermal wound healing. We present two approaches to modeling shrinking clusters. In the first approach, which is motivated by the Eden model, every point on the cluster periphery has equal chance of being healed. Noisy and noisefree versions of this model are investigated. In the second approach, DLA is employed in a unique way so that random walkers launched from infinity eventually reach the cluster and contribute to its reduction. Simulation results are presented which illustrate the evolution of the wound healing process for various wound shapes.     

Macroscopic models for fibroproliferative disorders: A review

In this paper we review some mathematical modelling of organ reparative processes (wound healing) for both the physiological and pathological case. The natural process of healing consists in a series of overlapping phases involving cells, chemicals, extracellular matrix (ECM) and the environment surrounding the wound site. Sometimes the healing process fails and the reparative mechanism produces pathological conditions which are commonly termed fibrosis or fibroproliferative disorders. Biological insight into the pathogenesis, progression and possible regression of fibrosis is lacking and many issues are still open. Mathematical modelling can surely play its part in this field and this paper is aimed at showing what has been done so far and what has still to be done to achieve a unified framework for studying these kinds of problems. Due to the high complexity of this phenomenon, multi-scale modelling is certainly the appropriate approach that should be used for studying these kinds of problems. Unfortunately most of the mathematical literature on this topic consists of macroscopic continuous models which fail to investigate processes occurring at smaller length scales (cellular, sub-cellular). We present a review of some of the mathematical literature, showing the widely used approaches, focusing on the interpretation of results and indicating possible developments in the study of these highly complex systems.     

A perturbation analysis of a mechanical model for stable spatial patterning in embryology

Summary We investigate a mechanical cell-traction mechanism that generates stationary spatial patterns. A linear analysis highlights the model's potential for these heterogeneous solutions. We use multiple-scale perturbation techniques to study the evolution of these solutions and compare our solutions with numerical simulations of the model system. We discuss some potential biological applications among which are the formation of ridge patterns, dermatoglyphs, and wound healing.     

On the construction of analytic solutions for a diffusion-reaction equation with a discontinuous switch mechanism

The existence of waiting times, before boundary motion sets in, for a diffusion-diffusion reaction equation with a discontinuous switch mechanism, is demonstrated. Limit cases of the waiting times are discussed in mathematical rigor. Further, analytic solutions for planar and circular wounds are derived. The waiting times, as predicted using these analytic solutions, are perfectly between the derived bounds. Furthermore, it is demonstrated by both physical reasoning and mathematical rigor that the movement of the boundary can be delayed once it starts moving. The proof of this assertion resides on continuity and monotonicity arguments. The theory sustains the construction of analytic solutions. The model is applied to simulation of biological processes with a threshold behavior, such as wound healing or tumor growth.     

Adaptive error control during gradient search for an elliptic optimization problem

In this article we describe a cost effective adaptive procedure for optimization of a quantity of interest of a solution of an elliptic problem with respect to parameters in the data, using a gradient search approach. The numerical error in both the quantity of interest and the computed gradient may affect the progression of the search algorithm, while the errors generally change at each step during the search algorithm. We address this by using an accurate a posteriori estimate for the error in a quantity of interest that indicates the effect of error on the computed gradient and so provides a measure for how to refine the discretization as the search proceeds. Specifically, we devise an adaptive algorithm to refine and unrefine the finite element mesh at each step in the search algorithm. We give basic examples and apply this technique to a model of a healing wound.     

基于弹性壳的三维群体细胞动力学模型

群体细胞迁移常见于胚胎发育、伤口愈合和肿瘤侵袭等各种生理和病理过程中,关于其动力学的研究对于揭示群体细胞迁移机理、深刻理解有关生物过程十分重要.该文构建了群体细胞的三维可变形壳状模型,提出了一种考虑细胞弹性形变和细胞间接触与黏附相互作用的群体细胞动力学理论,并发展了相应的数值算法.基于所发展的动力学模型与算法,对多细胞在嚢腔里的受限旋转运动进行了模拟,复现了相关实验现象,分析了细胞极性、细胞形变、胞间相互作用等因素对多细胞三维动力学的影响规律.     

The Elastodynamics of Embryonic Epidermal Wound Closure

This paper is concerned with the elastodynamics of embryonic epidermal wound closing. Underlying the recovery process for this type of wounds is a mechanism of wound recognition through directed cell-to-cell signaling. The observed actin filament realignment induced by the biological signals leads to a purse-string effect and the resulting (unknown) "active stresses." The circumferential contraction of the epidermis surrounding the wound is then determined by the laws of mechanics and propagation properties of the relevant cell–cell signaling that decays with distance. With the wound known to retract for a short period immediately after infliction, the quasi-equilibrium configuration reached during this initial phase serves as the initial condition for the dynamic wound closing phase. A small strain variation of the Murray–Sherratt model of the quasi-equilibrium problem will be formulated for speedy computation of this initial state at the inception of the wound closure phase, with the latter problem being the main concern of this paper. Some theoretical developments are found to be instrumental to an efficient algorithm for the otherwise time-consuming task of calculating the effect of the biological signals generated by the presence of a wound. Application of our elastodynamic model to the case of a circular wound suggests that the propagation range of our choice of cell–cell signaling mechanism must be above a certain minimum fraction of the wound radius for wound closure. As expected, stress concentration occurs adjacent to the edge of the remaining small wound near the end of the wound closing process. At that point, the present model is not expected to be adequate and more appropriate expressions of elastic strain and active stress induced by actin filaments may be in order. Other biological processes such as cell proliferation and differentiation may be involved.     

细胞趋硬性迁移的研究进展

细胞力学微环境可以调控许多细胞生理功能.特别地,在细胞力学微环境各种信号梯度的作用下,细胞可以定向地迁移.这些定向迁移可以显著影响伤口愈合、癌细胞转移和组织形貌发育等生理过程.目前为止,细胞的定向迁移主要包括:在化学药物梯度作用下的趋药性迁移,在黏附分子梯度作用下的趋触性迁移,以及在细胞外基质硬度梯度作用下的趋硬性迁移等.虽然细胞趋药性和趋触性迁移的力学机理得到了很好的研究,但是关于细胞趋硬性迁移的机制和作用还不清楚.该文重点介绍了细胞趋硬性的相关实验和理论研究进展,分析了不同研究间的联系与区别,讨论了细胞趋硬性迁移的潜在力学机制,提出尚存在的问题和未来可能的研究方向.     

Traveling waves in a coarse-grained model of volume-filling cell invasion: Simulations and comparisons

Many reaction–diffusion models produce traveling wave solutions that can be interpreted as waves of invasion in biological scenarios such as wound healing or tumor growth. These partial differential equation models have since been adapted to describe the interactions between cells and extracellular matrix (ECM), using a variety of different underlying assumptions. In this work, we derive a system of reaction–diffusion equations, with cross-species density-dependent diffusion, by coarse-graining an agent-based, volume-filling model of cell invasion into ECM. We study the resulting traveling wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic comparison between the behaviors observed in this model and those predicted by simpler models in the literature that do not take into account volume-filling effects in the same way. Our study justifies the use of some of these simpler, more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes, but it also reveals some interesting properties arising from the introduction of cell and ECM volume-filling effects, where standard model simplifications might not be appropriate.     

Mechanical properties of a carbon-fiber-reinforced plastic and the residual stresses in wound components made out of combined composites

A technique for calculating the residual stresses in wound components made out of mixed carbon-fiber-and glass-fiber-reinforced plastics and the results of the calculations are presented. The effect of the conditions under which the components are wound on the magnitude and nature of the residual stresses distribution in components made out of combined composites is investigated. The results of the calculations are compared with the experimental values.Translated from Mekhanika Polimerov, No. 6, pp. 996–1004, November–December, 1975.     

Experimental determination of residual stresses in wound unidirectional glass-reinforced plastics

The residual stresses in prestressed ring-shaped systems of wound unidirectional glass-reinforced tape have been investigated experimentally. The relation between the residual stress and the winding force has been established. A decrease in the specified prestress is demonstrated. The results of tensile tests on free rings of wound glass-reinforced tape are presented.Mekhanika Polimerov, Vol. 2, No. 1, pp. 123–129, 1966     

FE-Analysis of concentration dependent healing processes in a polymer matrix

In this contribution, a macroscopic four-phase model, based on the Theory of Porous Media, is presented to simulate healing processes in a polymer matrix which depend on the amount (concentration) of catalysts. Therefore, the healing process is described by the phase transition from liquid like healing agents to solid like healed material. This phase transition is a function depending on the concentration. To show the applicability of the developed model, a numerical example will be presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithmic strategy for the simulation of bone healing directly on computed tomography data

An algorithmic strategy for the modelling and simulation of bone healing is presented. The algorithm works directly on the computed tomography data and simulates, after an appropriate volume meshing, a mechainically driven healing concept which is based on competitive and dynamical mechanical parameters. The finite element simulations are done with realistic boundary conditions from patient-specific OpenSim simulations. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)